When does the Frullani type integral hold equality?

[mhill]

under what condition the does the equality hold ?

$$\int_{0}^{\infty} dt \frac{ f(at)-f(bt)}{g(t)}= (G(b)-G(a))(f(0)-f(\infty))$$

and $$\int dt g(t)$$

 

[HallsofIvy]

Did you mean to say
$$G(t)= \int dt g(t)$$
in your last line?

 

[mhill]

yes i meant the integral of g(t) ,i have been looking for generalizations of this integral but without any luck , under that condition can define g(t) and f(at) f(bt) so the integral above converge ?? thank

 

[HallsofIvy]

The generalized mean value theorem says that if f and g are differentiable on (a, b) and continuous on [a, b], then there exist c in (a, b) such that
$$\frac{f(b)- f(a)}{g(b)- g(a)}= \frac{f'(c)}{g'(c)}$$

That looks like an integral version to me.